Cartesian product

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ such that $a$ is an element of $A$ and $b$ is an element of $B$ . More generally, the Cartesian product of an ordered family of sets $A_1, A_2, \dotsc$ is the set $A_1 \times A_2 \times \dotsb$ of ordered tuples $(a_1, a_2, \dotsb)$ such that $a_j$ is an element of $A_j$ , for any positive integer $j$ for which we have specified a set $A_j$ .

Existence

Ordered Pairs

In the language of set theory, it is not trivial to define an ordered pair since the set $\{a,b\}$ and $\{b,a\}$ are equivalent. Thus, the definition of an ordered pair $(a,b)$ is the set $\{\{a\}, \{a,b\}\}$ Through this definition, the pair $(a,b)$ does not equal the pair $(b,a)$ since the set $\{\{a\}, \{a,b\}\}$ and $\{\{b\}, \{b,a\}\}$ are not equivalent. However, for the ordered pair $(a,a)$ the resulting set reduces to ${{a}}$ (do you see why?). Thus reversing the positions of $a$ in the ordered pair does not change the resulting set.

Generally, the ordered pair $(a, b, c, \dots )$ can be though of as nested ordered pairs: $(a, (b, (c, (\dots))))$ .

Page

Toolbox

Search

Cartesian product

Existence

Ordered Pairs

See Also